Question

If the sum of two unit vectors is a unit vector, then magnitude of difference is:

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\sqrt{5}$

Answer 1

The correct option is B $\sqrt{3}$

Let the unit vectors be $\overrightarrow{a}$ & $\overrightarrow{b}$
$\overrightarrow{R}$ be the resultant
$\to \overrightarrow{R}=\overrightarrow{a}+\overrightarrow{b}$ & ${\overrightarrow{R}}^{\prime }=\overrightarrow{a}-\overrightarrow{b}$
$|\overrightarrow{R}|=\sqrt{|a{|}^2+|b{|}^2+2|a||b|\cos \theta }$
$\Rightarrow 1=\sqrt{1+1+2\cos \theta }$
$\Rightarrow 1=2+2\cos \theta$
$\Rightarrow \cos \theta =-\frac{1}{2}$
$\Rightarrow \theta ={120}^o=\frac{2\pi }{3}$
$\therefore |R^{\prime }|=\sqrt{|a{|}^2+|b{|}^2+2|a||b|\cos (\pi -\theta )}=\sqrt{1+1+2\times 1/2}=\sqrt{3}$